More by Nikos I. Kavallaris

More by Takashi Suzuki

We're sorry, but we are unable to provide you with the
full text of this article because we are not able to identify you as
a subscriber. If you have a personal subscription to this journal,
then please login. If you are already logged in, then you may need
to update your profile to register your subscription. Read more about accessing full-text

Abstract

The non-local parabolic equation \[ v_t=\Delta v+\frac{\lambda e^v}{\int_\Omega
e^v}\quad\mbox{in $\Omega\times (0,T)$} \] associated with Dirichlet boundary and initial
conditions is considered here. This equation is a simplified version of the full
chemotaxis system. Let $\lambda^*$ be such that the corresponding steady-state problem has
no solutions for $\lambda>\lambda^*$, then it is expected that blow-up should occur in
this case. In fact, for $\lambda>\lambda^*$ and any bounded domain $\Omega\subset {\bf
R}^2$ it is proven, using Trudinger-Moser's inequality, that $\int_{\Omega}
e^{v(x,t)}dx\to \infty$ as $t\to T_{max}\leq \infty.$ Moreover, in this case, some
properties of the blow-up set are provided. For the two-dimensional radially symmetric
problem, i.e. when $\Omega=B(0,1),$ where it is known that $\lambda^*=8\,\pi,$ we prove
that $v$ blows up in finite time $T^* < \infty$ for $\lambda>8\,\pi$ and this blow-up
occurs only at the origin $r=0$ (single-point blow-up, mass concentration at the origin).